Find the ordered pair $(x,y)$ if

\begin{align*}
x+y&=(5-x)+(5-y),\\
x-y&=(x-1)+(y-1).
\end{align*}
Explanation: Adding the equations, we get

$$2x=8\Rightarrow x=4.$$Substituting this into the first equation, we get

$$4+y=1+5-y\Rightarrow y=1.$$Thus the ordered pair is $\boxed{(4,1)}$.